Developments in Structural Engineering and Construction Technology

Developments in Structural Engineering and Construction Technology

Editors: Djwantoro Hardjito Antoni Ima Muljati

Proceedings of the Benjamin Lumantarna Symposium on Structural Engineering and Construction Technology, Surabaya, Indonesia 14 September, 2012

Developments in Structural Engineering and Construction Technology Edited by Djwantoro Hardjito, Antoni & Ima Muljati Department of Civil Engineering Petra Christian University Surabaya, Indonesia

Cover design and lay-out by Deddi Duto Hartanto

Published by Institute for Research and Community Service Petra Christian University Surabaya, Indonesia

Developments in Structural Engineering and Construction Technology @2012, ISBN 978-979-99765-1-2

Copyright @2012 by Institute for Research and Community Service Petra Christian University, Surabaya, Indonesia All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by no means, electronic, mechanical, by photocopying, recording or otherwise, without written permission from the publisher. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: Institute for Research and Community Service Petra Christian University Jl. Siwalankerto 121-131, Surabaya 60236 Indonesia http://lppm.petra.ac.id

ISBN: 978-979-99765-1-2


PREFACE In recognition of Prof. Dr. Benjamin Lumantarna’s outstanding contribution to Structural Engineering, Construction Technology, and to his four decades service in Civil Engineering Education at Petra Christian University, a number of longtime friends, colleagues and former students from around the world were invited to present papers on the Benjamin Lumantarna Symposium on Structural Engineering and Construction Technology, a theme that has been a major part of his research, deep involvement and profound contributions. The symposium was held at Petra Christian University, Surabaya, Indonesia, on Friday, 14 September 2012; hosted by the Civil Engineering Department of the university in conjunction with the Golden Jubilee Celebration of the Department. We would like to thank the authors, the session chairmen and the sponsors for their contribution, which enable this symposium to be held. Our thanks also go to Civil Golden Jubilee Organizing Committee, Petra Christian University, for their hard work. As this publication represents a wealth of knowledge of eminent researchers and engineers involved in Structural Engineering, Construction Technology and Civil Engineering Education, it is hoped that this symposium volume will become a useful resource for the construction industry and Civil Engineering Education.

Djwantoro Hardjito, Antoni & Ima Muljati Petra Christian University, Surabaya, Indonesia September 2012


FOREWORD

The Benjamin Lumantarna Symposium is gladly presented as part of the Golden Jubilee of Civil Engineering Department, Petra Christian University. It would be a thanksgiving moment for The Lord’s guidance as well as a memorable moment to honor Professor Benjamin Lumantarna for his profound dedication along the years passed. Along those years, the development of Civil Engineering Department is closely related to the contributions shared by Professor Lumantarna who has dedicated himself for four decades in Petra Christian University. As the first professor of this university, he devotes himself not only for the improvement of the department, but also toward the development of Petra as higher education institution. All his thoughtful knowledge shared will be precious legacy for the next generation in many ways. Throughout this event, I would like to thank the organizing committee for their hard work in holding this event. I would also send my sincere gratitude to all contributors, editors and sponsors for their supports. It is hoped that this proceeding would be a valuable resource for the development of research, especially in civil engineering science and major.

Soli Deo Gloria!

Rolly Intan Rector Petra Christian University, Surabaya, Indonesia September 2012


THE BL I KNOW

Benjamin Lumantarna (BL) was born as Loo Djien Gie in Jombang, 26th November 1946. He is the youngest of five born from a great respected mother. As the youngest child, later he chose Benjamin as his new name. In 1972 he married Inge, a very understanding and supporting lady. Neal was born to them in 1974 and is the only son. Recently Nolan descended to the family. BL’s childhood, until his fifth grade, was spent in Sidoarjo. He later moved to Surabaya and finished the elementary and junior high school in St Joseph. He made a big change in his character building when he entered Petra high school instead of proceeding to the St Louis high school as St Joseph’s alumni usually did. At that time St Louis was still restricted for boys, whereas Petra accepted both genders. The once shy, timid, and basically introvert boy transformed into a more friendly, active, and dynamic lad. After finishing high school, in 1964 he signed up at Petra Christian University (PCU), and was enrolled in the Department of Civil Engineering. At that time PCU was still a young and unknown university, just three years established. He also applied to the Architectural Department of Surabaya Institute of Technology (ITS) and had joined the department for about two semesters before he abandoned it. BL was vivid in student organizations, external and internal. I remember that day, after the 1965 turmoil, when he led a demo to the consulate of People’s Republic of China as a protest to China’s involvement in the rebellion. He once acted as a chairman to the student council of PCU, and in various committees. Despite his activity in organizations, he never forgot his formal courses. He always ranked first, almost in every civil engineering course. That was due to his diligent and self-disciplined in study, but mostly that was caused by strong logic. He graduated in 1973 and was directly assigned as a faculty member in the department. In PCU he taught structural analysis which is the basis of civil engineering. Actually his debute as engineer had begun long before he graduated. In 1968 he had already applied as apprentice in a well-known engineering firm and had been positioned as the so called structural analist. That was his training camp to sharpen his ability in understanding design philosophy and also engineering construction problems.


In 1974 he entered the Asian Institute of Technology in Bangkok to pursue a master’s degree, and achieved it in due time. Returning to PCU, he transferred his knowledge to his students. He was the first to introduce computer programming, Basic and Fortran. At that time computers were rarely owned by private universities. To his students, he has contributed significantly to a better understanding of the behavior of structural materials. Basically he was interested in exact studies; thus, he did not like expertises like soil engineering which has many uncertainties. It was surprising that he later favors seismic design, which is very uncertain to most of us. In 1981 he went off to Canada where he entered University of Toronto to pursue a Ph.D., which he was able to achieve in 1984. Going back to PCU, he occupied the position as the head of General Administration Office. There he managed the administration system and did a lot of betterment, which is still perceived up to now. He is a keen and accurate reader, and should there be gaps or flaws in any regulations, he will be aware of it. In meetings, he is good at giving arguments and might prolong the session. However, some persons might consider him audacious. He and Gideon established an engineering consulting firm, i.e. Benjamin Gideon and Associates (BGA). This firm celebrated its silver jubilee in 2010. Under his lead, BGA has managed many building projects especially in the structural design. However, his ability is not confined in the structural design area; due to his strong logic and sharp technical intuition, he could manage disputes regarding other disciplines. He could not only find a discrepancy in an engineering data but also show how it should be done. With his BGA, he has attracted many young engineers, mostly alumni of PCU, to the field of structure engineering and inspired them with confidence to undertake various tasks. Now some of them have their own engineering company. He has written many publications and been active in many seminars, symposia, and other technical meetings. He gained the title of professor in 1991, and became the first professor in PCU. He was assigned as a reviewer or reader by the Directorate of Higher Education. Also, PCU assigned him as the chief editor for its journal, Dimensi Teknik Sipil. He later, in 2003, decided to change the journal to Civil Engineering Dimension (CED) to take a chance to become an international journal. His decision brings promising results. Until now, he is still holding the position of the chief editor of CED.


He should now be in his retiring days, but PCU still needs his service, and he is still in his capacity to serve. His firm, BGA is still blooming, and promises auspicious years to come.

Medio February 2012 Johannes I. Soewono A relative, friend, colleague, and ex-student


CONTENTS Page Preface

iv

Foreword

v

The BL I know J.I. Soewono

vi

A Robust Design Strategy for High-Tech Buildings S. L. Lee, C. G. Koh and D. K. H. Chua

1

A New Landscape of Civil Engineering Profession W. Kanok Nukulchai

12

Towards Rational Design Method for Strengthening by Overlay T. Ueda, D. Zhang and H. Furuuchi

14

Safe and Sustainable Tall Buildings-State of the Art P. Mendis

24

Generalized Fragility Relationships with Local Site Conditions for Probabilistic Performance-Based Seismic Risk Assessment of Bridge Inventories D.T. Lau, K. Vishnukanthan, C.L. Waller and S. Sivathayalan Application of Preloading to Improve the Load Capacity of Foundation S. Prawono Performance Based Seismic Procedure using Direct Displacement Design for Sheathed Cold-Formed Steel (SCFS) Structures G. H. Kusuma

34

52

60

Application of Queuing Theory in Construction Management K. Teknomo

76

Global Multidisciplinary Learning in Construction Education R. Soetanto and M. Childs

89

Structural Strengthening using Carbon Fiber Reinforced Polymer Hartono

104

Limitations in Simplified Approach in Assessing Performance of Façade under Blast Pressures 111 R. Lumantarna, T. Ngo and P. Mendis


Seismic Assessment of Structures in Regions of Low to Moderate Seismicity E. Lumantarna, N. Lam and J. Wilson

127

Effect of Live Load on Seismic Response of a Horizontally Curved Bridge H. Wibowo, D. M. Sanford, I. G. Buckle and D. H. Sanders

147

Integrating Emotional Intelligence, Political Skill, and Transformational Leadership in Construction R.Y. Sunindijo

162

Learning from Local Wisdom: Friction Damper in Traditional Building B. Lumantarna and P. Pudjisuryadi

176

Application of Queuing Theory in Construction Management K. Teknomo Ateneo de Manila University, Philippines

Abstract: The purpose of this paper is to present a tutorial on how to apply queuing theory in construction management. Two case studies on concreting and earth moving illustrate how we model the construction activities as queuing systems systematically. Through the numerical examples, it was shown that when the customer cost is much lower than the server cost, queuing system can be simplified only to incorporate the constraint equation. The queueing constraint equation is suggested to be used as engineering rule of thumb. Keywords: Rule of thumb, constraint equation, concreting, earth moving

Introduction Queuing is traditionally defined as a process where people, materials or information need to wait at certain time to get a service. Queuing theory has been developed since the beginning of last century (by A. K. Erlang in 1917). Despite all the advancement of the queuing theory in almost a century, many people still have lack of understanding about how to manage queue. In our daily life, we still face a lot of queuing from school, to bank and from restaurant to toilet. We still are being frustrated with our daily traffic and a long queue in registration and security check. In fact, the queue does not diminish by the advancement of our knowledge in queuing theory but it grows into more complex reasons. Public and business understanding of queuing is still very far from the science. For instance, instead of distributing the queue over time and space, many managers do the very common mistake by making policy to concentrate the demand over time and space. Instead of solving the queuing problem, they create more severe queues. The lack of understanding on how to solve queueing problems requires public education. One best part of public education is to educate the engineers. As this forum is an excellent opportunity to give talk to Civil Engineers, especially Construction Managers, I would like to point out how we can use queueing theory in practice of construction field. The purpose of this paper is to present a tutorial on how to apply queuing theory in construction management. Through classical case studies of concreting and earth moving activities, I will like to encourage Civil Engineers and Construction Managers to use the vast knowledge of queueing theory that have been developed by mathematicians and management scientists for about a century. It should be noted that this paper is not the first paper which discuss the applications of queuing theory in construction management. In fact, there are a few books and

papers [1,2] had discussed about this topic in general. A study by Cheng et al [3] develops a construction management process reengineering performance measurement model by applying queuing theory to calculate process operation time in order to strike an optimal balance between process execution demand and manpower service capacity. Unlike these literatures, in this paper, I describe systematically how to model the construction activities into a queuing system. After showing the formulas, I also extend the treatment of M/M/1 or M/M/s queuing model in the existing books and papers into economic analysis and finally to come out with a simple engineering rule of thumb. To understand queueing theory fully, one need to have background in statistics and differential equations and to be able to manipulate Markov chain. Looking at the recent development of vacation queuing theory (example: Tian and Zhang in 2006 [4]) and queuing network (example: Yue et al in 2009 [5]), the trend of queuing theory development is toward more precision with requires higher mathematical manipulation. While higher precision queuing theory computation is great for the body of knowledge, this trend also produces wider gap between the theory and practice. Hall [6] cited an argument that operation research profession could and should be more scientific and less mathematical. We should concern with how the system behave and less concern with abstract symbol manipulation. Therefore, the presentation of the case studies in this paper is more qualitative than quantitative. All the formulas are either common sense or well known in the queuing theory that their proofs can be traced back in the references. Construction practitioners and engineers love simplification and rule of thumb. It would be shown at the end of this paper that we can simplify our treatment of queuing system into an engineering rule of thumb.

CASE STUDY 1: CONCRETING In this first case study, we are looking at concreting activity as a queuing system. Unlike queuing in supermarket where there is a single server to serve one customer, concreting operation requires many agents such as crane, hoist or bucket or concrete pump, placement crew, and vibrating crew as well as concrete trucks. The clutters in the agents may become the source of misunderstanding on how to model concreting activity as a queuing system. Which agents will play role as servers and which agent should play role as customers in the queuing system? What kind of optimization we would like to model through the queuing system? How many servers should we provide?. Two main components of a queuing system are customers and servers. Customer is person or thing that demands for service. Customer does not have to be a person and does not necessarily have to wait for service. Servers provide service to the customers. In the case of concreting activities, which agents are the servers and which agents are the customers? My suggestion to solve this problem is to use a simple technique as the following. First, we identify stakeholders and then we find out

the flow of activity in which we say that every type of agents is a server for their immediate customers. After that, we take side with the stakeholder and identify the most expensive agents as the servers. Let us identify the two stakeholders: contractor and concrete company. From the contractor point of view, the concrete trucks are the customers and what they provide, the crew and crane with bucket or the concrete pumps, are the servers. From the concrete company’s point of view, the concrete trucks are the servers while all the concrete pumps equipments and placement crew from the contractor’s side are the customers. For the sake of uniformity, let us take side with the contractor point of view. The other point of view would be equivalent anyway. Now we can look at the flow of the concreting activities and say that the crane and bucket (or the concrete pump or hoist) is the server for the concrete trucks. The placement and vibrating crew are the servers for the crane and bucket. Using this serial system, as we identify the most expensive agent, the concrete pumps should be identified as the servers for both the crew and the concrete trucks.

Figure 1. Concreting Activities as a Serial Queuing System. Concrete pump is the server for both systems.

Queuing theory is rich with optimization. Our next problem is to answer what kind of optimization we would like to model through the queuing system? In our concreting activities, the placement and vibrating crew are usually the ordinary workers that always available on the construction site. These servers must operate together to serve single customers of the concrete truck. If one of the servers is not available, the service of concreting cannot be done. Thus, normally we do not want to optimize the number of the crew. In fact, the number of placement crew is directly related to the number of concrete pumps or the hoists or the cranes. Say, to place and vibrate ν cubic meter of concrete per hour, we need x number of crew. If each concrete pump will provide g cubic meter of concrete per hour, we can easily find the number of required total crew to serve the n concrete pumps as:

g ⋅x Xn = n    v 

(1)

The notation ⋅ is ceiling function to get the lowest integer that higher than the argument. The unit time can be set as either hour or minute as convenient of the modeler. For uniformity, we use hourly unit of time. The direct relation between the number of crew and the number of concrete pumps simplifies our queuing system. Instead of having a serial queues, we can now combine the placement crew with their equipments (such as concrete pump, or crane with bucket or hoist) as one unit. We would like to optimize the number of servers, which is the number of concrete pumps (or cranes or hoists). Our optimization problem becomes how many concrete pumps (and eventually the crew) we would like to provide such that we can accomplish the concreting activities at minimum cost. Let us assume, just at the moment, that the number of crew and the number of concrete trucks that can be hired are unlimited and the space for the concrete trucks and concrete pumps are also unlimited. Later we will visit and release these assumptions.

Figure 2. Concreting Activities as a Simple Queuing System.

If we provide too many concrete pumps, we may accomplish the concreting activities faster but it is also at higher cost of renting the concrete pumps and hiring the crew. In other words, by adding the number of servers, the queuing system incurs higher server cost. If we provide too few concrete pumps, we may think that the total cost of the queuing system will be lower due to lower cost. However, when we provide less number of servers, the delay of the concrete trucks will be more than necessary. If the waiting time of the concrete trucks is too long, the concrete will be hardened and the overall concreting activities will be delayed and the overall cost will be even higher. Thus, at less number of servers, the queuing system incurs higher customers cost. As we think in term of system, the total cost of queuing system must include both server side and customer side. Optimal situation happens when both concrete trucks and concrete pumps would be minimal. Let us give notation

cs

to a

constant unit server cost which includes the renting of one unit of concrete pump

together with hiring cost of the crew to serve one concrete pump. Notation

cc

indicates a unit customer cost function which includes the waiting cost of one concrete truck per hour. It should be noted that

cc is a function of time rather than

a constant because the unit cost is higher significantly when the waiting time is longer. Since the behavior of this function over time is gradual increase at low waiting time and sudden increase until infinity at higher waiting time (due to hardening of the concrete), an exponential function with shape parameter beta will serve the purpose of this function. Parameter alpha is used to scale the value of the cost linearly.

cc = α exp(β t )

(2)

To find the value of the parameters, we need to use non-linear regression. Taking the natural logarithm of both side of Equation 2 produces a simple linear regression equation ln cc

= ln α + β t . If we have at least two points to calibrate

the regression, the parameter values can be easily found. For instance, we have range of time t in hour. We would consider the customer cost to be very high (say, $10,000 or more) if the waiting time is more than 3 hours due to risk of hardening of the concrete. If the waiting time is 30 minutes, the customer cost would be $1000. Inputting points (0.5, $1000) and (3.0, $10,000) into the linear regression equation produces α = 630.96 and β = 0.92. .

Figure 3. Exponential function of unit customer cost

Then, the total cost that would be minimized is computed as

C = n ⋅ cs + W ⋅ cc

(3)

While we can set schedule for the concrete trucks to arrive at certain regular interval, in practice however due to traffic condition, usually the arrival of the concrete trucks will be stochastic with inter arrival rate at the schedule time. Similarly, since the crew workability is now part of the queuing system, the service time to pour into cast and vibrate certain unit volume of concrete is also stochastic in nature. Given the input of average and variation of inter arrival time and the service time to place a unit volume of concrete; we can compute the value of average waiting time W . The model to compute the waiting time will be discussed in the next sections of this paper. For each number of server n , we compute the total cost C based on equation (3). The optimum number of server is the one that minimize the total cost.

CASE STUDY 2: EARTH MOVING Carmichaela [7] explained the application of queuing theory for earth moving. The paper discussed the assumptions on the service discipline, on steady-state behavior and on the probability distributions for the service times. My treatment here is more qualitative in nature. In this second case study, we see earth moving activity as a queuing system. Earth moving activities have several agents: the excavators, the dump trucks and the loader (i.e. bulldozer). On the source of the quarry, the excavators cut the top soil and fill into the dump truck. The dump trucks then bring the soils to the construction site, dump the soils and the loader are ready to spread the soils to fill or to create the landscape of the land. We are faced with the same simple questions again: how to model earth moving activity as a queuing system? Which agents are the servers and which agent are the customers? What kind of optimization we would like to model through the queuing system? How many servers should we provide? First, let us attempt to answer how to model earth moving activity as a queuing system. In earth moving activity, we may have three separate stakeholders: the quarry owner which operates the excavator, the transportation company which operates the dump truck and the contractor who operates the loader. From each stakeholder points of view, they may think that they provide the service to the customer. The quarry owner may think the excavator is the server and the dump truck is the customer. The trucking company will think that the dump truck is the server to serve the excavator and the loader. Similarly, the contractor may think the loader is the server to the dump truck.

It is also possible that the three stakeholders are actually one company. In this case, we will have system point of view. Looking at the flow of the earth moving activity, we have a serial queueing system Excavators Dump trucks Loaders.

Figure 4. Earth moving Activities as a Serial Queuing System. Dump truck is the customer for both queuing systems.

If the distance between quarry and the construction site is relatively big that the traffic conditions may affect the order of arrival of the dump trucks, we may treat the serial queuing system as two separate queuing systems. First queuing system happens in the quarry where excavators are the server and the dump trucks are the customers. The dump trucks are waiting for the excavators to be filled. The second queuing system happens in the construction site where the loaders are the server to the dump trucks. The trucks are waiting for the loader to clear the filled soil before it can dump the soil to the next slot. The large distance assumption between quarry and the construction site will greatly simplify our queuing systems. From this point of view, when we identify the most expensive agents as the servers (in term of rental fee per hour), it reveals that the dump trucks are the customers for both queuing systems.

Figure 5. Earth Moving Activities as Two Simple Queuing Systems

To answer what kind of optimization we would like to model through the queuing system, let us assume for the moment that we can hire can hire unlimited dump

truck and the space for the trucks to queue for both excavators in the quarry and loader in the construction site are also unlimited. Similar to the previous case study, we will visit and release these assumptions later. Since both queuing system in the quarry and in the construction site has similar characteristics, for simplicity of the explanation, only the quarry site will be discussed. If we provide too many excavators, we can finish the earth moving activity faster at higher cost of renting the excavators. Clearly, providing higher number of servers incurs higher server cost. On the other hand, providing too few excavators will create long queue for the trucks to wait and the overall earth moving activities will be delayed. Eventually, in this case the overall cost will be higher. Thus, providing lower number of servers incurs higher customers cost. With similar reasoning to the first case study of concreting, we can use Equation 2 and Equation 3 to find the optimum number of servers (that is the number of excavators or the number of loaders) that will minimize the total cost of the queuing system. Note in contrast that in the literatures (such as Carmichaela [7]) exist what is called Griffis’ Application of Queuing Theory to determine the number of trucks to perform earth moving activities. In this paper, we will not consider this application.

QUEUING MODELS Having the two case studies of concreting and earth moving, in this section we would like to have an integrated treatment of the two case studies. Even though the agents of the two case studies are different, they can be abstracted simply as servers and customers. Having this abstraction, we can now treat them as a simple queuing system. The servers are characterized by service time distribution. In a simplified queuing theory, we can have either stochastic or deterministic service time. When we have stochastic service time and the variation of the service time is equal or almost equal to the average service time, we say that the service time is having or approaching Markovian distribution. In this case, the service time distribution fits into exponential distribution with mean

µ is equal to the variance σ s 2 .

The customers are characterized by arrival distribution. When we have stochastic arrival distribution and the variation is equal or almost equal to the arrival rate, we say that arrival having or approaching Markovian distribution. In this case, the discrete arrival distribution fits into Poisson distribution with mean λ is equal to the variance

σ a −2 . Note in this case, the inter arrival rate λ −1 is simply

an inverse of the arrival rate and the distribution of inter arrival rate would be exponential distribution.

The actual distributions of service time and arrival should be gathered and fit into the closest theoretical statistical distribution and then the appropriate formulas for queuing theory will be used to predict the performance of the queuing system. If the appropriate formulas of the queuing theory for the proper distribution are not available, we need to build simulation model. In practice, however, we often want to simplify this process because building a specific simulation model may require some cost on itself. When the mean is equal to variance for both distributions of service time and arrival, we can use most often used queuing formulas. The Kendal notation of such queuing system would be M/M/s where the first letter is to indicate arrival distribution and the second letter is to indicate the service time distribution and the third letter represent the number of servers. For this type of queuing system, we can compute the performance of the queuing system in term of the average number of customer in the system (in waiting line and being served) as

L = Lq + ρ =

P0 ρ s +1

( s − 1)!( s − ρ )

2

+ρ

(4)

Where,

 s −1 ρ i ρ s  s µ   P0 =  ∑ +    i =0 i ! s !  s µ − λ  

−1

(5)

is defined as probability that there are no customers in the system. The ratio of arrival rate and service rate is given as

ρ=

λ µ

(6)

The average time a customer spends in the system in waiting line and being served is computed using Little’s Law

W=

L

(7)

λ

Now if the mean is not equal to the variance, we need more general type of queuing system, which Kendal notation would be G/G/s. Unfortunately, the formula of the queuing performance for general type queuing system does not exist yet. Only the approximation of G/G/s queuing system is available (Allen & Cunneen’s as cited by Hall [6]). Given the coefficient of variation for inter-arrival time

ca and

coefficient of variation for service time

cs ,

average customers in waiting line waiting for service as:

we can compute the

 c 2 + cs2  Lq (G / G / s ) = Lq ( M / M / s ) ⋅  a   2 

(8)

The average queue length follows the first part of Equation 4 and the average waiting time follows the Little’s law in Equation 7. Earlier in the first case study, we have assumed unrealistically that the space for the concrete trucks and concrete pumps are also unlimited. Similarly, in the second case study, it was assumed unrealistically that the space for the dump trucks to queue for both excavators in the quarry and loader in the construction site are also unlimited. We can release these assumptions by setting the limiting capacity on the number of customers that can be accommodated in the queuing system. The Kendal notation of this type of queuing system is M/M/s/N where the last letter indicates the capacity of customers that can enter the queuing system. This type of queuing also means that the trucks that are not allowed to enter the quarry or construction site when the space capacity has been reached. Another unrealistic assumption involves unlimited number of placement & vibrating crew, the number of concrete trucks or dump truck that can be hired. We can release this kind of assumption on the size of the calling source by limiting the number of customers. The Kendal notation of this type of queuing system is M/M/s/N/N where the last letter indicates the size of the customers. With the removal of unrealistic assumptions above, we have discussed fully on how to model the examples of construction management case studies into queueing system. The next section will describe simple analysis based on the formulas above.

QUEUING ANALYSIS In this section, I will give numerical illustration of applying the queuing formulas above for the first case study of concreting. The second case study bears similar technique and therefore shall not be repeated. In this analysis, we would like to answer two questions on What is the typical schedule of the concrete trucks such that the concrete pumps will not get idle 90% of the time? What is the optimum number of server? To make the problem more quantitative, the typical values are given as follow. Suppose we would like to cast concrete of one floor of a building with volume of 1500 cubic meters. As typical concrete truck can carry about 6 cubic meter, the activity requires about 250 trucks. A concrete pump has typical capacity of g = 50 cubic meters per hour. Thus, one concrete pump can serve about µ = 8 concrete trucks/hour. The concreting activity of one floor may require 30 concrete pumps to be finished in an hour, or 30 hours using only single concrete pump.

Renting cost of a concrete pump is $150/hour and labor cost to place and vibrate 50 cubic meters of concrete in an hour is about $100/hour. This gives unit server cost of

cs = $250/hour. The unit customer cost follows the previous explanation

of using exponential function of Equation 2. To answer the first question of truck schedule, we use Equations 5 and 6 by inputting µ = 8 concrete trucks/hour for various number of server s and customer arrival rate λ such that the idle probability

P0 is less than or equal to 10%. It

should be noted that within queuing theory, there is a constraint that if the utilization of the queuing system must not be larger than one. If the utilization is larger than one, the queue length and waiting time would be at infinity.

U=

ρ s

=

λ